The electrostatic potential energy between proton and electron separated by a distance 1 Å is

Options

A. 13.6 eV
B. 27.2 eV
C. `-14.4 eV`
D. 1.44 eV

Answer

To find the electrostatic potential energy (U) between a proton and an electron separated by a distance of 1 Å (angstrom), we can use the formula for electrostatic potential energy:

\[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]

where:
- \( U \) is the electrostatic potential energy,
- \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \),
- \( q_1 \) and \( q_2 \) are the charges of the proton and electron, respectively,
- \( r \) is the separation distance between the charges.

### Step 1: Identify the charges
The charge of a proton (\( q_1 \)) is \( +1.6 \times 10^{-19} \, \text{C} \) and the charge of an electron (\( q_2 \)) is \( -1.6 \times 10^{-19} \, \text{C} \).

### Step 2: Convert the distance
The distance \( r \) is given as 1 Å, which is equal to \( 1 \times 10^{-10} \, \text{m} \).

### Step 3: Substitute the values into the formula
Now, substituting the values into the formula:

\[ U = \frac{(9 \times 10^9) \cdot (1.6 \times 10^{-19}) \cdot (-1.6 \times 10^{-19})}{1 \times 10^{-10}} \]

### Step 4: Calculate the numerator
Calculating the numerator:

\[ 9 \times 10^9 \cdot 1.6 \times 10^{-19} \cdot (-1.6 \times 10^{-19}) = 9 \times 10^9 \cdot (-2.56 \times 10^{-38}) = -23.04 \times 10^{-29} \]

### Step 5: Divide by the distance
Now, divide by \( r \):

\[ U = \frac{-23.04 \times 10^{-29}}{1 \times 10^{-10}} = -23.04 \times 10^{-19} \, \text{J} \]

### Step 6: Convert to electron volts
To convert joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):

\[ U = \frac{-23.04 \times 10^{-19}}{1.6 \times 10^{-19}} \approx -14.4 \, \text{eV} \]

### Final Answer
The electrostatic potential energy between a proton and an electron separated by a distance of 1 Å is approximately \( -14.4 \, \text{eV} \).

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